This research statement summarizes Susovan Pal's postdoctoral research in two areas: 1) Regularity and asymptotic conformality of quasiconformal minimal Lagrangian diffeomorphic extensions of quasisymmetric circle homeomorphisms. This focuses on proving these extensions are asymptotically conformal if the boundary maps are symmetric. 2) Discrete geometry of left conformally natural homeomorphisms of the unit disk from a discrete viewpoint. This constructs homeomorphisms between polygons in the disk that preserve a weighted minimal distance property. The goal is to show these homeomorphisms converge to a continuous one.

1. POSTDOCTORAL RESEARCH STATEMENT
SUSOVAN PAL
This research has received funding from the European Research Council under
the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC
grant agreement no
FP7-246918
1. Overview
My postdoctoral research has been in two areas related to Riemann surfaces and
differential geometry:
My present research is in:
. Regularity and asymptotic conformality of quasiconformal minimal Lagrangian
diffeomorphic extensions of quasisymmetric circle homeomorphisms, using tools
from AdS -geometry.
. Discrete geometry: Left conformally natural homeomorphism of unit disk from
discrete viewpoint.
2. Asymptotic conformality of quasiconformal minimal Lagrangian
extensions
One of my current research slightly deviates from my Ph.D. work and focuses
on regularity and asymptotic conformality of quasiconformal minimal Lagrangian
diffeomorphic extensions of quasisymmetric circle homeomorphisms. A minimal
Lagrangian diffeomorphism between two Riemann surfaces is one which is area-
preserving and its graph is a minimal surface in the product of the two Riemann
surfaces. We have the following key theorems:
Theorem 1. [14]:Let φ : S1
→ S1
be a quasisymmetric homeomorphism. Then
there exists a unique quasiconformal minimal Lagrangian diffeomorphism Φ : D →
D such that ∂Φ = φ.
The Beurling Ahlfors extension and the Douady-Earle extensions, previously
mentioned, have the properties that if the boundary homeomorphisms are sym-
metric (i.e. the constant of quasisymmetry is almost 1 when restricted to small
intervals), then the corresponding extensions are asymptotically conformal, i.e. be-
haves almost like a conformal map outside a big compact subset of D. We are hoping
that the same property will be reflected by the minimal Lagrangian extensions. For
the relevant notations, please consult [14]
Theorem 2. [14]: Let Γu be an acausal C0,1
-graph in ∂∞AdS3
∗
. If Γ does not
contain any lightlike segments, then there exists a unique maximal(mean curvature
zero), spacelike surface bounding Γ, i.e. then there exists a unique maximal space-
like surface Su such that ∂Su = Γu.
Date: March 9, 2015.
1

2. 2 SUSOVAN PAL
It is interesting to see how the width w(K = Γu) depends on the quasisymmetric
constant of u : S1
→ S1
.
Conjecture 1. Let u : S1
→ S1
be a k-quasisymmetric homeomorphism and let
w = w(Γu) be the width of its graph. Then w → 0ask → 0.
The above conjecture is motivated by the fact that in the AdS3=model for example,
if we take P0 = H×{0}, then P0 is the restriction of the Identity map Id : S1
→ S1
.
and the width of the graph of the Identity map is zero.
Given a spacelike surface S in AdS∗
3 , we consider [14] ΦS = Φr ◦ Φ−1
l : P0 → P0,
and BS : T (S) → T (S).
Theorem 3. ΦS is quasiconformal ⇔ the eigenvalues of BS are in (-1,1)⇔ the
width of ∂∞S < π/2.
So in brief, we have:
u : S1
→ S1
is quasisymmetric ⇒ Γu ⊂ S1
× S1
AdS∗
3 is an acausal C0,1
-graph
with width < π/2 ⇒ the unique maximal surface S with ∂∞S = Γu has its shape
operator BS with eigenvalues in (−1, 1) ⇒ ΦS : P0 → P0 with ∂ΦS = φ is quasi-
conformal.
We conjecture the following:
Conjecture 2. Let ΦS, as defined above be K-quasiconformal, and let its eigen-
values be {δK, −δK}. Then δK → 0 if and only if K → 1.
Theorem 4. Let width w = w(∂∞S) < π/2, and the eigenvalues of BS are
{δw, −δw}. Then δw → 0 as w → 0.
The above conjectures, if true prove that the minimal Lagrangin extension of a
symmetric homeomorphism is asymptotically conformal.
We also have an unrelated conjecture, but similar to the one we proved for
Douady-Earle extensions [8]:
Conjecture 3. If u is C1
, then the corresponding Φl, Φr [14] are C1
on D, hence
ΦS, its minimal Lagrangian extension is C1
on D.
3. Left conformally natural homeomorphism of unit disk from
discrete viewpoint
This is another of my ongoing research. Although the problem can be stated
in higher dimensions, we will restrict ourselves working with two dimensions for
now: let P1, P2, ...Pn be n points on S1
, n ≥ 3. Given any z ∈ D, the (hyperbolic)
distance from z to any of the Pj is infinite. However, we can still define the ”signed
distance” [17] as follows: consider any horocycle h in D, and define:

3. POSTDOCTORAL RESEARCH STATEMENT 3
δ(z, h) = δh(z) =



-dD(z, h) if z is inside h
0 ifz ∈ h
dD(z, h) if z is outside h
Now, let wj, 1 ≤ j ≤ n be n positive numbers. It can be seen that Σn
j=1wjδhj
(z) →
∞ as z → ∂D = S1
. So, as in [17], we can define a point of ’weighted minimal
distance sum’ in D. It can be shown exactly as in [17] that:
Theorem 5. The origin is a point of weighted minimal distance sum Σn
j=1wjδhj
(.) =
Σn
j=1wjδ(., hj) from the horospheres at P1, P2, ...Pn if and only if Σn
j=1wjPj = 0
Because of the above algebraic condition in lemma 11, the point of weighted mini-
mal distance sum is unique. Next, we take Pj = e
2π.i(j−1)
n , 1 ≤ j ≤ n. Let f be an
orientation-preserving homeomorphism of the unit circle and let Qj = f(Pj). Let
P, Q denote the interiors of the polygon bounded by the corresponding hyperbolic
geodesics. Let z ∈ P, and Tz(w) = w−z
1−¯zw . Then the tangent lines through Tz(Pj)
form a Euclidean polygon, whose side lengths we call wj(z). Note by Stokes’ theo-
rem: Σn
j=1wj(z)Pj = 0, which by lemma 9 above, imply z is the point of minimal
distance sum with weights wj(z). In symbols, z = argminΣn
j=1wj(z)δ(., hPj
)
Now, define Fn,f : P → Q by: Fn,f (z) = argminΣn
j=1wj(z).Qj = argminΣn
j=1wj(z).f(Pj),
i.e. F carries the point of minimal distance sum from Pj with weights wj(z) to the
one with same weights, but from Qj. The following property of Fn,f have been
already proved as a part of this project:
Lemma 1. Fn,f is a local diffeomorphism for any f.
Fn,α◦f = α ◦ Fn,f ∀α ∈ Aut(D), where Aut(D) is the set of conformal automor-
phisms of D.
The proofs of the second one is more or less a routine verification and the first one
follows after application of implicit function theorem on the smooth map (z, w, f) →
Σn
j=1wj(z)Twf(Pj) and using lemma 11 above.
We conjecture that:
Conjecture 4. Fn,f is a diffeomorphism from P = P(n) → Q(n), and letting
n → ∞ so that P(n) ”converge to D by filling it up from inside”, Fn,f will converge
to a homeomorphism Ff of D satisfying Fα◦f = α ◦ Ff ∀α ∈ Aut(D).
References
[1] W. Abikoff, ‘Conformal barycenters and the Douady-Earle extension - A discrete dynamical
approach’, Jour. d’Analyse Math. 86 (2002) 221-234.
[2] L. V. Ahlfors, Lectures on Quasiconformal Mapping, Van Nostrand Mathematical Studies 10
(Van Nostrand-Reinhold, Princeton, N. J., 1966).
[3] A. Beurling and L. V. Ahlfors, ‘The boundary correspondence for quasiconformal mappings’,
Acta Math. 96 (1956) 125-142.
[4] Peter Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhauser, pp. 213-215.
[5] Issac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, Chapter 1, 11.

4. 4 SUSOVAN PAL
[6] A. Douady and C. J. Earle, ‘Conformally natural extension of homeomorphisms of circle’,
Acta Math. 157 (1986) 23-48.
[7] C. J. Earle, ‘Angular derivatives of the barycentric extension’, Complex Variables 11 (1989)
189-195.
[8] Jun Hu and Susovan Pal, ‘Boundary differentiablity of Douady-Earle extensions of diffeomor-
phims of Sn’, accepted at the Pure and Applied Mathematics Quarterly.
[9] ——, Douady-Earle extensions of C1,α circle diffeomorphims, in preparation.
[10] O. Lehto and K. I. Virtanen, Quasiconformal Mapping (Springer-Verlag, New York, Berlin,
1973).
[11] Susovan Pal, Construction of a Closed Hyperbolic Surface of arbitrarily small eigenvalues of
prescribed serial numbers,to appear in the Proceedings of the Ahlfors-Bers colloquium VI in
2011.
[12] Burton Randol, Small Eigenvalues of the Laplace Operator on Compact Riemann Surfaces,
Bulletin of the American Mathematical Society 80 (1974) 996-1000.
[13] Dennis Sullivan, Bounds, quadratic differentials and renormalizations conjecture, Mathemat-
ics into the twenty-first century, Vol. 2, Providence, RI, AMS
[14] F. Bonsante, J-M Schlenker, Maximal surface and the Universal Teichmuller space, Invent
math (2010) 182: 279333, DOI 10.1007/s00222-010-0263-x
[15] F. Labourie, Surfaces convexes dans lespace hyperbolique et CP1-structures, J. Lond. Math.
Soc., II. Ser. 45, 549565 (1992)
[16] R. SCoen, The role of harmonic mappings in rigidity and deformation problems, Lecture Notes
in Pure and Appl. Math., vol. 143, pp. 179200. Dekker, New York (1993). MR MR1201611
(94g:58055)
[17] B. Springborn, A Unique representation of polygedral types. Centering visa Mobius transfor-
mations Mathematische Zeitschrift 249, 513-517 (2005)
Etage 3, 80, boulevard DIDEROT, 75012 PARIS, France
E-mail address: susovan97@gmail.com